Finiteness properties of centralisers in Out(FN) and automorphisms of free-by-cyclic groups

Let F be a finitely generated free group. We prove that the centraliser of any element of Out(F) or Aut(F) is of type VF. We also prove that the (outer) automorphism group of any free-by-cyclic group M=F_φZ is of type VF, and so are the McCool subgroups of Out(M). This relies on the construction of a new Out(M)-invariant splitting of M, that enables us to reduce the proof of the above statements to the case where the outer class of $φ$ has finite order.